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Rejection Based Face Detection

Michael Elad*

Scientific Computing and Computational Mathematics

Stanford University

The Computer Vision Workshop

March 18th, 2002

* Collaboration with Y. Hel-Or and R. Keshet

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Part 1 Part 1

Introduction of the Introduction of the Problem and Problem and

Basic ConsiderationsBasic Considerations

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1. The Task

Face (target) Detector

Part 1

Input image

Comments:

1. Extension to color

2. Faces vs. general targets

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2. Requirements

Detect frontal & vertical faces: All spatial position, all scales Any person, any expression Robust to illumination conditions Old/young, man/woman, hair, glasses.

Design Goals: Fast algorithm Accurate (False Positive / False Negative)

Part 1

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3. Frontal-Vertical Faces

Taken from the ORL Database

Part 1

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Face Finder

Suspected

Face Positions

Input Image

Classifier

Draw L*L blocks from

each location

4. Scale-Position

and in each resolution

layer

Compose a pyramid with 1:f resolution ratio (f=1.2)

Part 1

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5. Classifier Design

A classifier is a parametric (J parameters) function C(Z,θ) of the

form }1,1{:,ZC JL2

Q1: What parametric form to use? Linear or non-linear? What kind of non-linear? Etc.

Q2: Having chosen the parametric form, how do we find appropriate set of parameters θ ?

Need to answer two questions:

Part 1

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6. Algorithm Complexity

Searching faces in a given scale, for a 1000 by 2000 pixels image, the classifier is applied 2e6 times

(Q1) Choosing the parametric form: keep in mind that the algorithm’s complexity is governed by the classifier complexity

Part 1

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7. Training by Examples xN

1kkX xy NN1kkY

Part 1

1,YC,Nk1

1,XC,Nk1

kY

kX

(Q2) Finding Suitable Parameters:

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8. Geometric Interpretation

{ } XNk k 1

X=

{ } YNk k 1

Y=

C(Z,θ) is to drawing a separating manifold between the two classes

+1 -1

2L

Part 1

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Part 2 Part 2

SOMESOME Previous WorkPrevious Work

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1. Neural Networks Choose C(Z,θ) to be a Neural Network

(NN). Add prior knowledge in order to:

Control the structure of the net, Choose the proper kind (RBF ?), Pre-condition the data (clustering)

Representative Previous Work: Juel & March (1996), and Rowley & Kanade (1998), and Sung & Poggio (1998).

NN leads to a Complex Classifier

Part 2

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2. Support Vector Machine

Choose C(Z,θ) to be a based on SVM. Add prior knowledge in order to:

• Prune the support vectors,

• Choose the proper kind (RBF, Polynomial ?),

• Pre-condition the data (clustering)

Representative Previous Work:• Osuna, Freund, & Girosi (1997),

• Bassiou et.al.(1998),

• Terrillon et. al. (2000). SVM leads to a Complex Classifier

Part 2

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3. Rejection Based Build C(Z,θ) as a combination of

weak (simple to design and activate) classifiers.

Apply the weak classifiers sequentially while rejecting non-faces.

Representative Previous Work: Rowley & Kanade (1998) Elad, Hel-Or, & Keshet (1998), Amit, Geman & Jedyank (1998), Osdachi, Gotsman & Keren (2001), and Viola & Jones (2001).

Part 2

Fast (and accurate) classifier

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Input Blocks

4. The Rejection Idea

Detected

Rejected

Weak Classifie

r # n

…

Weak Classifie

r # 2

Rejected

Weak Classifie

r # 3

Reje

ctedWeak

Classifier # 4

Rejected

Classifier

Part 2

Weak Classifie

r # 1

Rejected

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5. Supporting Theory

Rejection – Nayar & Baker (1995) - Application of rejection while applying the sequence of weak classifiers.

Part 2

(Ada) Boosting – Freund & Schapire (1990-2000) – Using a group of weak classifiers in order to design a successful complex classifier. Decision-Tree – Tree structured classification (the rejection approach here is a simple dyadic tree).

Maximal Rejection – Elad, Hel-Or & Keshet (1998) – Greedy approach towards rejection.

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Part 3 Part 3

Maximal Rejection Maximal Rejection ClassificationClassification

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1. Linear Classification (LC)

We propose LC as our weak classifier:

0TZsign,ZC

Part 3

+1

-1

{ } XNk k 1

X=

{ } YNk k 1

Y=

2LHyperplane

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Find θ1 and two decision levels such that the number of rejected non-faces is maximized

while finding all faces

1 2 1d ,d

2. Maximal Rejection

1d

2d

Projected onto θ1

Part 3

Non-Faces

Faces XN

k k 1X

YNk k 1

Y

Rejected non-faces

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Projected onto θ1

Taking ONLY the remaining non-faces:Find θ2 and two decision levels such that the number of rejected non-faces is maximized

while finding all faces

3. Iterations

1 2 2d ,d

Projected onto θ2

1d

2d

Rejected

points

Part 3

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4. Maximizing Rejection

XNk k 1

X Maximal Rejection

Maximal distance between these two

PDF’s

We need a measure for this distance which will

be appropriate and easy to use

YNk k 1

Y

Part 3

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5. One Sided Distance

This distance is asymmetric !! It describes the average distance between points of Y to the X-PDF,

PX().

Define a distance between a point and a PDF by

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1 0 x x2x

2 20 x x

2x

D ,P P dr

m r

r

xP

0

2 2 2

x y x y2 x y 1 y 2

x

(m m ) r rD P ,P D ,Px( ) P ( )d

r

Part 3

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6. Final Measure

2 2 2 2 2 2

x y x y x y x y3 x y 2 2

x y

(m m ) r r (m m ) r rD P ,P P(Y) P(X)

r r

In the case of face detection in images we have

P(X)<<P(Y)

Part 3

We Should Maximize

(GEP)

TT

X YX Y X Y

TX

M M M M R Rf

R

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Y X

X X

N N 2T Tk j

j 1k 1N N 2T T

k jj 1k 1

X Y

fX X

Maximize the following function:

7. Different Method 1

Maximize the distance between all the pairs of

[face, non-face]

Minimize the distance between all the pairs of

[face, face]

The same Expression

T

TC

R

Q

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If the two PDF’s are assumed Gaussians, their KL distance is

given by

2 2 2

x y x yKL x y 2

x

x

y

(m m ) r rD P ,P

2r

rln 1

r

And we get a similar expression

XNk k 1

X

YNk k 1

Y

8. Different Method 2

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9. LimitationsPart 3

* More accurately, if in the convex hull of the face set there are non-faces

The discriminated zone is a parallelogram. Thus, if the faces set is non-convex*, zero false alarm discrimination is impossible – Solution: Second layer.

Even if the faces-set is convex, convergence to zero false-alarms is not guaranteed. – Solution: Clustering the non-faces.

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8. Convexity? Can we assume that the Faces set is convex?

- We are dealing with frontal and vertical faces only

- We are dealing with a low-resolution representation of the faces

- Are they any non-faces that are convex combination of faces ?

Part 3

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Chapter 4 Chapter 4

Results & ConclusionsResults & Conclusions

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Kernels for finding faces (15·15) and eyes (7·15).

Searching for eyes and faces sequentially - very efficient!

Face DB: 204 images of 40 people (ORL-DB after some screening). Each image is also rotated 5 and vertically flipped - to produce 1224 Face images.

Non-Face DB: 54 images - All the possible positions in all resolution layers and vertically flipped - about 40·106 non-face images.

Core MRC applied (no second layer, no clustering).

1. Details Part 4

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2. Results - 1

Out of 44 faces, 10 faces are undetected, and 1 false alarm(the undetected faces are circled - they are either rotated or strongly

shadowed)

Part 4

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All faces detected with no false alarms

3. Results - 2Part 4

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4. Results - 3

All faces detected with 1 false alarm(looking closer, this false alarm can be considered

as face)

Part 4

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5. More Details A set of 15 kernels - the first typically removes

about 90% of the pixels from further consideration. Other kernels give a rejection of 50%.

The algorithm requires slightly more that one convolution of the image (per each resolution layer).

Compared to state-of-the-art results: Accuracy – Similar to (slightly inferior in FA) to Rowley

and Viola. Speed – Similar to Viola – much faster (factor of ~10)

compared to Rowley.

Part 4

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6 .Conclusions

Rejection-based classification - effective and accurate.

Basic idea – group of weak classifiers applied sequentially followed each by rejection decision.

Theory – Boosting, Decision tree, Rejection based classification, and MRC.

The Maximal-Rejection Classification (MRC): Fast – in close to one convolution we get face detection, Simple – easy to train, apply, debug, maintain, and extend. Modular – to match hardware/time constraints. Limitations – Can be overcome.

More details – http://www-sccm.stanford.edu/~elad

Part 4

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7 . More Topics1. Why scale-invariant measure?2. How we got the final distance expression? 3. Relation of the MRC to Fisher Linear Discriminant4. Structure of the algorithm5. Number of convolutions per pixel6. Using color7. Extending to 2D rotated faces8. Extension to 3D rotated faces9. Relevancy to target detection10. Additional ingredients for better performance

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1. Scale-Invariant

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1 0 x x2x

2 20 x x

2x

D ,P P dr

m r

r

xP

0

Same distance for

xP

0

xP

0

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TT

X YX Y X Y

TX

M M M M R Rf

R

In this expression:1. The two classes means are encouraged to

get far from each other 2. The Y-class is encouraged to spread as

much as possible, and 3. The X-class is encouraged to condense to a

near-constant valueThus, getting good rejection performance.

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2. The Distance Expression

Nk k 1Z

Nk

k 1N T

k kk 1

1M Z

N

1Z M Z M

N

R

T T Tk kz Z m M, r R

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xT

yT

xyxyT

2x

2y

Txyxy

MMMM

r

rmmmm

R

R

back

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3. Relation to FLD*

*FLD - Fisher Linear Discriminant

Assume that

and

Gaussians

XNk k 1

X

YNk k 1

Y

Minimize variances

Maximize mean difference

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Nk k 1Z

Nk

k 1N T

k kk 1

1M Z

N

1Z M Z M

N

R

T T Tk kz Z m M, r R

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2T T

X Y

T TX Y

M Mf

R R

Maximize

Minimize

TXM T

YM

TXR T

YR

TTX Y X Y

TX Y

M M M M

R R

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2y

2y

2x

2yx

2x

2y

2x

2yx

r

rrmm)X(P

r

rrmm)Y(P

In the MRC we got the expression for the distance

The distance of the Y points to the X-

distribution

The distance of the X points to the Y-

distribution

If P(X)=P(Y)=0.5 we maximize

2y

2y

2x

2yx

2x

2y

2x

2yx

r

rrmm

r

rrmm

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Instead of maximizing the sum

2y

2y

2x

2yx

2x

2y

2x

2yx

r

rrmm

r

rrmm

Minimize the inverse of the two expressions (the inverse represent the

proximity)

2yx

2y

2x

2y

2x

2yx

2y

2y

2x

2yx

2x

mm

rrMin

rrmm

r

rrmm

rMin

back

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1 2 j,d ,d

T1k

T2k

Y d

k or

Y d

Remove

Sub-set YN (j 1)j 1k k 1

Y

YN (j) Threshold? END

4. Algorithm Structure

XNk k 1

X Compute

X X,MR

YN (0)0k

k 1Y

Compute

Y Y,MR

Minimize f(θ)

& find thresholds

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Is value in1 2 j

d ,d

No more Kernels Fac

e

Yes

No

Non Face

j j 1

Project onto the

next Kernel

J1 2 j 1

,d ,d

back

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5. Counting Convolutions

6.08.1

9.02.1

99.01~

235.0k112k

1k

• Assume that the first kernel rejection is 0<<1 (I.e. of the incoming blocks are rejected).

• Assume also that the other stages rejection rate is 0.5.

• Then, the number of overall convolutions per pixel is given by

back

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6. Using Color

back

Several options:

Trivial approach – use the same algorithm with blocks of L-by-L by 3.

Exploit color redundancy – work in HSV space with decimated versions of the Hue and the Saturation layers.

Rejection approach – Design a (possibly non-spatial) color-based simple classifier and use it as the first stage rejection.

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7. 2D-Rotated Faces

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Frontal &

Vertical Face

Detector

Pose Estimatio

n and Alignment

Input block

Face/Non-Face

Remarks:

1. A set of rotated kernels can be used instead of actually rotating the input block

2. Estimating the pose can be done with a relatively simple system (few convolutions).

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8. 3D-Rotated Faces

back

A possible solution:

1. Cluster the face-set to same-view angle faces and design a Final classifier for each group using the rejection approach

2. Apply a pre-classifier for fast rejection at the beginning of the process.

3. Apply a mid-classifier to map to the appropriate cluster with the suitable angle

Mid-clas. For

Angle

Crude Rejection

Input block

Face/Non-Face

Final Stage

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9. Faces vs. Targets

back

Treating other targets can be done using the same concepts of

Treatment of scale and location

Building and training sets

Designing a rejection based approach (e.g. MRC)

Boosting the resulting classifier

The specific characteristics of the target in mind could be exploited to fine-tune and improve the above general tools.

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10. Further Improvements

back

• Pre-processing – linear kind does not cost

• Regularization – compensating for shortage in examples

• Boosted training – enrich the non-face group by finding false-alarms and train on those again

• Boosted classifier – Use the sequence of weak-classifier outputs and apply yet another classifier on them –use ada-boosting or simple additional linear classifier

• Constrained linear classifiers for simpler classifier

• Can apply kernel methods to extend to non-linear version